A key factor that controls the BCS-BEC crossover in FeSe\(_{1-x}\) S\(_x\)  is the multiband character. As shown previously [1]}, [2]}, in FeSe the strong interband interaction between electron and hole pockets prevents the splitting of the \(T_{\rm pair}\)  and the \(T_c\) , determined by the superfluid stiffness. We note that this strong interband interaction is also an important ingredient in the orbital-selective scenario [3]}, [4]}, [5]}, [6]} and therefore we expect that even taking orbital selectivity into account, the main result, i.e. mean-field like jump of the specific heat will remain the same. At the same time, much less is known on the evolution of the ratio of the interband versus intraband interactions upon approaching \(x_c\)  in FeSe\(_{1-x}\) S\(_x\) . Furthermore, recent experimental [7]} and theoretical [8]}, [9]} works pointed out an interesting possibility of the substantial non-local nematic order on the \(d_{xy}\)  orbital and/or inter-orbital-nematicity between \(d_{xy}\)  and \(d_{yz}\)  orbitals in FeSe. The most important consequence of this phenomenon would be an additional Lifshitz transition in FeSe\(_{1-x}\) S\(_x\)  upon S substitution, which occurs close to \(x_c\) . This in turn implies the appearance of the incipient band near \(x_c\)  (see for example Fig.6 in Ref.[8]}). On general grounds, the presence of the incipient band would enhance the tendency towards the BCS-BEC crossover. In addition, given nearly equal mixture of the \(s\) - and \(d\) -wave symmetry components of the superconducting gap in FeSe [11]} to express such anisotropic gap, it is natural to assume that they are competing in the tetragonal phase of FeSe\(_{1-x}\) S\(_x\) . In particular, the ratio of the pairing interactions in the \(d\) -wave and \(s\) -wave channels \(|g^d_{\rm ee}|/g^s_{\rm eh}\)  may, in addition to the incipient band, be another key parameter of the BCS-BEC crossover as follows from our model calculations (see Fig.S4). The pair-formation temperature \(T_{\rm pair}\)  and condensation temperature \(T_{\rm c}\)  can be split more with increasing the ratio of intraband and interband interactions \(|g^d_{\rm ee}|/g^s_{\rm eh}\)  (see Fig.S4(b)) without changing \(\varepsilon _{\rm F}\) . This quantitatively explains the appearance of strong coupling superconductivity for \(x>x_c\)  with no apparent enhancement of \(\Delta /\varepsilon _F\) . A remarkable and unexpected feature is the absence of pseudogap above \(T_{\rm c}\)  in the STS spectra, despite the presence of giant superconducting fluctuations in thermodynamic quantities. It should be noted that the relationship between the preformed pairs and the pseudogap formation is still elusive even in the ultra-cold atoms [12]}. In addition, the onset temperature of the pseudogap may differ from that of fluctuations [13]}. Moreover, it has been pointed out that the pseudogap phenomena in multiband systems are markedly altered from those in the single band systems. For instance, in multiband systems, the pseudogap formation is less prominent by the reduction of its onset temperature [14]}. Quite recently, the BEC-like dispersion of the Bogoliubov quasiparticles has been reported in the tetragonal FeSe\(_{1-x}\) S\(_x\)  by using ARPES measurements [15]}. Although the direct comparison of the ARPES data with the present thermodynamic results is not straightforward, because ARPES measures only the portion of hole Fermi pocket, it provides spectroscopic evidence of the BCS-BEC crossover. The ARPES measurements also report the presence of distinct pseudogap in the hole pocket. However, we point out that this is not an apparent contradiction to the present STM results. It has been suggested that the DOS of electron pocket is much larger than that of hole pockets. As STM measures the DOS integrated over all bands, STM is insensitive to the pseudogap formation in the hole pocket. Therefore, the STM spectra, combined with the ARPES results, suggest that the pseudogap formation is also orbital dependent.
